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A363550 Number of partitions of [n] having exactly one parity change within the partition.

%I A363550 #14 Jun 09 2023 19:05:28
%S A363550 0,0,2,1,3,2,7,5,20,15,67,52,255,203,1080,877,5017,4140,25287,21147,
%T A363550 137122,115975,794545,678570,4892167,4213597,31858034,27644437,
%U A363550 218543759,190899322,1573857867,1382958545,11863100692,10480142147,93345011951,82864869804
%N A363550 Number of partitions of [n] having exactly one parity change within the partition.
%C A363550 The blocks are ordered with increasing least elements.
%H A363550 Alois P. Heinz, <a href="/A363550/b363550.txt">Table of n, a(n) for n = 0..1153</a>
%H A363550 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A363550 a(0) = a(1) = 0, for n>=2: a(n) = A000110((n-1)/2) if n is odd, a(n) = A011968(n/2) if n is even.
%F A363550 a(2*n) = a(2*n-1) + a(2*n+1) for n>=2.
%e A363550 a(2) = 2: 12, 1|2.
%e A363550 a(3) = 1: 13|2.
%e A363550 a(4) = 3: 134|2, 13|24, 13|2|4.
%e A363550 a(5) = 2: 135|24, 135|2|4.
%e A363550 a(6) = 7: 1356|24, 135|246, 135|24|6, 1356|2|4, 135|26|4, 135|2|46, 135|2|4|6.
%e A363550 a(7) = 5: 1357|246, 1357|24|6, 1357|26|4, 1357|2|46, 1357|2|4|6.
%e A363550 a(8) = 20: 13578|246, 1357|2468, 1357|246|8, 13578|24|6, 1357|248|6, 1357|24|68, 1357|24|6|8, 13578|26|4, 1357|268|4, 1357|26|48, 1357|26|4|8, 13578|2|46, 1357|28|46, 1357|2|468, 1357|2|46|8, 13578|2|4|6, 1357|28|4|6, 1357|2|48|6, 1357|2|4|68, 1357|2|4|6|8.
%p A363550 b:= proc(n) option remember; `if`(n=0, 1,
%p A363550       add(b(n-j)*binomial(n-1, j-1), j=1..n))
%p A363550     end:
%p A363550 a:= n-> `if`(n<2, 0, (h-> b(h)+`if`(n::even, b(h-1), 0))(iquo(n, 2))):
%p A363550 seq(a(n), n=0..35);
%Y A363550 Column k=1 of A363519.
%Y A363550 Cf. A000110, A011968.
%K A363550 nonn
%O A363550 0,3
%A A363550 _Alois P. Heinz_, Jun 09 2023